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(1)Solving Linear Inequalities Chapter Overview and Pacing. PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. 1. 1. 0.5. 0.5. 2 (with 6-2 Preview). 1. 1 (with 6-2 Preview). 0.5. Solving Multi-Step Inequalities (pp. 332–337) • Solve linear inequalities involving more that one operation. • Solve linear inequalities involving the Distributive Property.. 1. 1. 1. 0.5. Solving Compound Inequalities (pp. 339–344) • Solve compound inequalities containing the word and and graph their solution sets. • Solve compound inequalities containing the word or and graph their solution sets.. 2. 2. 1. 1. Solving Open Sentences Involving Absolute Value (pp. 345–351) • Solve absolute value equations. • Solve absolute value inequalities.. 2. 2. 1. 1. Graphing Inequalities in Two Variables (pp. 352–358) • Graph inequalities on the coordinate plane. • Solve real-world problems involving linear inequalities. Follow-Up: Use a graphing calculator to investigate graphs of inequalities.. 2. 2 (with 6-6 Follow-Up). 1. 1.5 (with 6-6 Follow-Up). Study Guide and Practice Test (pp. 359–363) Standardized Test Practice (pp. 364–365). 1. 1. 1 (with 6-6 Follow-Up). 0.5. Chapter Assessment. 1. 1. 0.5. 0.5. 12. 11. 7. 6. Solving Inequalities by Addition and Subtraction (pp. 318–323) • Solve linear inequalities by using addition. • Solve linear inequalities by using subtraction. Solving Inequalities by Multiplication and Division (pp. 324–331) Preview: Use algebra tiles to solve inequalities. • Solve linear inequalities by using multiplication. • Solve linear inequalities by using division.. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 316A. Chapter 6. Solving Linear Inequalities.

(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 343–344. 345–346. 347. 348. 349–350. 351–352. 353. 354. 393. 355–356. 357–358. 359. 360. 393, 395. 361–362. 363–364. 365. 366. 367–368. 369–370. 371. 372. 394. 373–374. 375–376. 377. 378. 394. 79–80, 83–84. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 6 RESOURCE MASTERS. See pages T12–T13.. Materials. SC 11. 46. 6-1. 6-1. SC 12. 47. 6-2. 6-2. 48. 6-3. 6-3. 14. 49. 6-4. 6-4. 15. GCS 33. 50. 6-5. 6-5. stopwatch. GCS 34. 51. 6-6. 6-6. (Follow-Up: graphing calculator). 379–392, 396–398. (Preview: algebra tiles, equation mat, self-adhesive notes) graphing calculator. 52. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 6. Solving Linear Inequalities. 316B.

(3) Mathematical Connections and Background Continuity of Instruction Prior Knowledge Students solved open sentence inequalities in Chapter 1. Chapter 2 had them graphing rational numbers and exploring absolute value. Solving single-step and multi-step equations was developed in Chapter 3. Students learned to graph linear equations in Chapter 4.. This Chapter Students develop the properties for solving inequalities. They apply the Addition and Subtraction Properties of Inequalities to solve inequalities. They also use the Multiplication and Division Properties of Inequalities, where the sign is sometimes reversed, to solve inequalities. They solve single-step and multistep inequalities. Students solve compound inequalities, as well as equations and inequalities that contain absolute values. The chapter ends with graphing inequalities in two variables.. Future Connections In future math studies, students solve and graph inequalities of other types of functions, such as quadratics. They also apply solving and graphing inequalities in Algebra 2 to linear programming where the maximum profit for a situation is determined. Inequalities are used in many biological areas, such as determining appropriate parameters for populations of species in various regions.. 316C. Chapter 6. Solving Linear Inequalities. Solving Inequalities by Addition and Subtraction To solve an equation, isolate the variable so that it has a coefficient of 1 on one side of the equal sign. An inequality is solved the same way. The Addition Property of Inequality is used in the same way as the Addition Property of Equality. It states that any number can be added to each side of an inequality and the result is a true inequality. The same is true for the Subtraction Property of Inequality: a number can be subtracted from each side of an inequality and the result is a true inequality. There are infinitely many solutions to an inequality. The solutions to inequalities can be written in set builder notation, for example {x | x 3). This is read as the set of all numbers x such that x is greater than 3. The number found when solving an inequality is a boundary that is sometimes included in the solution and sometimes not. It is included in the solution if the inequality sign is or , but it is not included if the symbol is or . If the boundary number is included, a solid dot is placed at that point on the number line. If the number is not included, use an open circle. Then draw an arrow to the right if the rest of the solution set is greater than the boundary, or to the left if the rest of the solution set is less than the boundary.. Solving Inequalities by Multiplication and Division Inequalities that include multiplication or division of the variable can also be solved. The same principles as found in the Multiplication and Division Properties of Equality are used, with one main difference. If an inequality is multiplied or divided by the same negative number on each side, the inequality symbol is reversed. The symbol must be reversed to result in a true inequality. The inequality sign is not reversed if each side is multiplied or divided by the same positive number. You multiply or divide by a negative number only if the coefficient of the variable is negative.. Solving Multi-Step Inequalities Solve multi-step inequalities using the same process as for solving multi-step equations. Work backward using inverse operations to undo the operations. After each side is simplified using the Distributive Property and/or combining like terms, work in the opposite order of the order of operations. The Addition and Subtraction Properties of Inequality are applied first, followed by the Multiplication and Division Properties of Inequalities..

(4) If the solution is an untrue statement, such as 4 8, there is no solution. If the solution results in a statement that is always true, such as 5 3, then the solution is the set of all real numbers. A solution can always be checked by substituting it back into the inequality.. Solving Compound Inequalities In a compound inequality, one variable is related to two different amounts with two inequality signs. The signs may be the same or they may be different. If and is written between the inequalities, or the variable expression is between the two inequality signs, the graph is the intersection of the two inequalities. This is because the solution must be true for both inequalities. If or is written between the two inequalities, the graph is the union of the two inequalities. This is because the solution can be true for either inequality.. Graphing Inequalities in Two Variables The solution set of an inequality, like that of an equation, is all ordered pairs that make the statement true. Similar to the solution set of an equation in two variables, the solution set of an inequality in two variables is graphed on a coordinate plane. However, the solution set of an inequality is not linear. It does have a linear boundary, but it covers a region called a half-plane. First graph the inequality as if it contained an equal sign like an equation. This is the boundary line. If the inequality is or , then the line is dashed. A solid line is graphed for and . These relate to the circle and dot on a number line. Select a point in either half-plane and test it in the inequality. (0, 0) is a good point to use if it is not on the boundary line. If the resulting statement is true, shade the half-plane that contains the point. If the statement is false, shade the other half plane.. Solving Open Sentences Involving Absolute Value An absolute value open sentence can be an equation or an inequality. The value inside the absolute value symbols could be positive or negative. The absolute value represents the distance a number is from zero on a number line. Absolute value equations can be solved by graphing them or by writing them as a compound sentence and solving algebraically. To solve algebraically, write the expression inside the symbol as equal to the given value and then equal to the opposite of the given value. Solve each equation. Write both solutions inside one set of brackets. An absolute value inequality is written as a compound inequality. If the absolute value is on the left and the inequality symbol is or , the compound sentence is written with and. If the absolute value is on the left and the inequality symbol is or , the compound sentence is written with or. To solve the first case, write the expression from inside the absolute value symbol, the or , and the value to the right of the sign. Then write the expression, the opposite inequality sign, and the opposite value. Solve both inequalities and write the solution set as an intersection. To solve the second case you follow the same process, only write the solution set as a union using or before solving both inequalities.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Solving Multi-Step Inequalities (Lesson 15) • Solving Compound Inequalities (Lesson 16) • Graphing Inequalities in Two Variables (Lesson 17) Chapter 6. Solving Linear Inequalities. 316D.

(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Technology/Internet. Ongoing. Prerequisite Skills, pp. 317, 323, 331, 337, 344, 351 Practice Quiz 1, p. 331 Practice Quiz 2, p. 344. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 79–80, 83–84 Quizzes, CRM pp. 393–394 Mid-Chapter Test, CRM p. 395 Study Guide and Intervention, CRM pp. 343–344, 349–350, 355–356, 361–362, 367–368, 373–374. AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Mixed Review. pp. 323, 331, 337, 344, 351, 357. Cumulative Review, CRM p. 396. Error Analysis. Find the Error, pp. 329, 348 Common Misconceptions, p. 326. Find the Error, TWE pp. 329, 348 Unlocking Misconceptions, TWE pp. 321, 334 Tips for New Teachers, TWE pp. 323, 334. Standardized Test Practice. pp. 323, 328, 329, 331, 337, 343, 351, 357, 363, 364–365. TWE pp. 364–365 Standardized Test Practice, CRM pp. 397–398. Open-Ended Assessment. Writing in Math, pp. 323, 331, 337, 343, 351, 357 Open Ended, pp. 321, 328, 334, 341, 348, 355 Standardized Test, p. 365. Modeling: TWE pp. 323, 351 Speaking: TWE pp. 337, 357 Writing: TWE pp. 331, 344 Open-Ended Assessment, CRM p. 391. Chapter Assessment. Study Guide, pp. 359–362 Practice Test, p. 363. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 379–384 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 385–390 Vocabulary Test/Review, CRM p. 392. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 316E. Chapter 6. Solving Linear Inequalities.

(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 6-3. 14 Solving Inequalities. 6-4. 15 Writing Inequalities. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 46–52 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 317 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 321, 328, 334, 341, 348, 355) • Reading Mathematics, p. 338 • Writing in Math questions in every lesson, pp. 323, 331, 337, 343, 351, 357 • Reading Study Tip, pp. 319, 339, 340 • WebQuest, p. 357 Teacher Wraparound Edition • Foldables Study Organizer, pp. 317, 359 • Study Notebook suggestions, pp. 321, 324, 328, 334, 338, 341, 348, 355 • Modeling activities, pp. 323, 351 • Speaking activities, pp. 337, 357 • Writing activities, pp. 331, 344 • Differentiated Instruction, (Verbal/Linguistic), p. 320 • ELL Resources, pp. 316, 320, 322, 330, 336, 338, 343, 350, 356, 359 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 6 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 6 Resource Masters, pp. 347, 353, 359, 365, 371, 377) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 6.4, 6.6, 6.8. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 6. Solving Linear Inequalities. 316F.

(7) Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. Solving Linear Inequalities • Lessons 6-1 through 6-3 Solve linear inequalities. • Lesson 6-4 Solve compound inequalities and graph their solution sets. • Lesson 6-5 Solve absolute value equations and inequalities. • Lesson 6-6 Graph inequalities in the coordinate plane.. Key Vocabulary • • • • •. set-builder notation (p. 319) compound inequality (p. 339) intersection (p. 339) union (p. 340) half-plane (p. 353). Inequalities are used to represent various real-world situations in which a quantity must fall within a range of possible values. For example, figure skaters and gymnasts frequently want to know what they need to score to win a competition. That score can be represented by an inequality. You will learn how a competitor can determine what score is needed to win in Lesson 6-1.. Lesson 6-1 6-2 Preview 6-2 6-3 6-4 6-5 6-6 6-6 Follow-Up. NCTM Standards. Local Objectives. 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 4, 6, 8, 9, 10 2, 4, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 316. Chapter 6 Solving Linear Inequalities. 316 Chapter 6 Solving Linear Inequalities. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 6 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 6 test..

(8) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 6. For Lessons 6-1 and 6-3. Solve Equations. Solve each equation. (For review, see Lessons 3-2, 3-4, and 3-5.) 6. 19 1. t 31 84 53. 2. b 17 23 40. 2 1 1 4. d 1 3 2 6 8. 2a 5 3a 4 1. 3. 18 27 f 9. 5. 3r 45 4r 45 6. 5m 7 4m 12 7. 3y 4 16 4 1 9. k 4 7 22. n1 2. 10. 4.3b 1.8 8.25 11. 6s 12 2(s 2) 12. n 3 7. 2. 1.5. 4. For Lesson 6-5. Evaluate Absolute Values. Find each value. (For review, see Lesson 2-1.) 13. 8 8. 14. 20 20. 15. 30 30. 16. 1.5 1.5. 17. 14 7 7. 18. 1 16 15. 19. 2 3 1. 20. 7 10 3. For Lesson 6-6. Graph Equations with Two Variables. This section provides a review of the basic concepts needed before beginning Chapter 6. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 79–80 and 83–84. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. For Lesson 6-2. Graph each equation. (For review, see Lesson 4-5.) 21–28. See pp. 365A–365D. 21. 2x 2y 6. 22. x 3y 3. 23. y 2x 3. 24. y 4. 1 25. x y 2. 26. 3x 6 2y. 27. 15 3(x y). 28. 2 x 2y. 6-3 6-4 6-5 6-6. Make this Foldable to record information about solving linear inequalities. Begin with two sheets of notebook paper. Fold and Cut Fold in half along the width. Cut along fold from edges to margin.. Fold. Prerequisite Skill Multiplication and division equations (p. 323) Multi-step equations (p. 331) Graphing integers on a number line (p. 337) Absolute values (p. 344) Graphing linear equations (p. 351). Fold a New Paper and Cut Fold in half along the width. Cut along fold between margins.. Label. Insert first sheet through second sheet and align folds.. Label each page with a lesson number and title.. Solving Linear Inequalities. As you read and study the chapter, fill the journal with notes, diagrams, and examples of linear inequalities.. Reading and Writing. Chapter 6. Solving Linear Inequalities 317. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data and Journal Writing After students make their Foldable journals, have them label each page to correspond to a lesson in the chapter. Students can use their Foldables to take notes, record concepts, and define terms. They can also use them to record the direction and progress of learning, to describe positive and negative experiences during learning, to write about personal associations and experiences, and to list examples of ways in which new knowledge has or will be used in their daily life. Chapter 6 Solving Linear Inequalities. 317.

(9) Solving Inequalities by Addition and Subtraction. Lesson Notes. 1 Focus 5-Minute Check Transparency 6-1 Use as a quiz or a review of Chapter 5.. • Solve linear inequalities by using addition. • Solve linear inequalities by using subtraction.. Vocabulary. In the 1999–2000 school year, more high schools offered girls’ track and field than girls’ volleyball.. Girls gear up for high school sports High school girls are playing sports in record numbers, almost 2.7 million in the 1999-2000 school year. Most popular girls sports by number of schools offering each program:. 14,587 13,426. 16,526 Basketball. If 20 schools added girls’ track and field and 20 schools added girls’ volleyball the next school year, there would still be more schools offering girls’ track and field than schools offering girls’ volleyball.. Building on Prior Knowledge. are inequalities used to describe school sports? Ask students: • Is the number of schools that offer volleyball greater than or less than the number of schools that offer track and field? less than • Suppose 1200 schools added track and field, and 1200 added volleyball. Would there be more schools offering track and field, or volleyball? track and field • Sports The kinds of sports offered in schools often reflects the sports that are popular in the region. Which sports are offered at your school? Which have the most participants?. are inequalities used to describe school sports?. • set-builder notation. Mathematical Background notes are available for this lesson on p. 316C.. In Chapter 3, students learned to solve equations using addition and subtraction. In this lesson, they should recognize that solving inequalities by addition and subtraction is a very similar process.. USA TODAY Snapshots®. 14,587 20. ?. 13,426 20. Source: National Federation of State High School Associations. 14,607 13,446. Study Tip Look Back. 14,587 Track and field 13,426 Volleyball 13,009 11,277 Softball Cross country. By Ellen J. Horrow and Alejandro Gonzalez, USA TODAY. SOLVE INEQUALITIES BY ADDITION Recall that statements with greater than (), less than (), greater than or equal to ( ), or less than or equal to () are inequalities. The sports application illustrates the Addition Property of Inequalities.. To review inequalities, see Lesson 1-3.. Addition Property of Inequalities • Words. If any number is added to each side of a true inequality, the resulting inequality is also true.. • Symbols For all numbers a, b, and c, the. • Example. following are true. 1. If a b, then a c b c. 2. If a b, then a c b c. This property is also true when and are replaced with. 27 2676 8 13. and .. Example 1 Solve by Adding Solve t 45 13. Then check your solution. Original inequality t 45 13 t 45 45 13 45 Add 45 to each side. t 58 This means all numbers less than or equal to 58. CHECK Substitute 58, a number less than 58, and a number greater than 58. Let t 58. Let t 50. Let t 60. ? ? ? 50 45 13 60 45 13 58 45 13 5 13 ⻫ 15 13 13 13 ⻫ The solution is the set {all numbers less than or equal to 58}. 318. Chapter 6 Solving Linear Inequalities. Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters • Study Guide and Intervention, pp. 343–344 • Skills Practice, p. 345 • Practice, p. 346 • Reading to Learn Mathematics, p. 347 • Enrichment, p. 348. Parent and Student Study Guide Workbook, p. 46 School-to-Career Masters, p. 11. Transparencies 5-Minute Check Transparency 6-1 Answer Key Transparencies. Technology Interactive Chalkboard.

(10) Study Tip Reading Math. {t|t 58} is read the set of all numbers t such that t is less than or equal to 58.. The solution of the inequality in Example 1 was expressed as a set. A more concise way of writing a solution set is to use set-builder notation . The solution in set-builder notation is {tt 58}. The solution to Example 1 can also be represented on a number line. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. The heavy arrow pointing to the left shows that the inequality includes all numbers less than 58.. 58. 59. 60. 61. 62. 63. SOLVE INEQUALITIES BY ADDITION. In-Class Examples. The dot at 58 shows that 58 is included in the inequality.. Power Point®. 1 Solve s 12 65. Then. check your solution. s 77 or {all numbers greater than 77}. Example 2 Graph the Solution. y 9. Then graph it on a number line. {y | y 21}. 2 Solve 12. Solve 7 x 4. Then graph it on a number line. 7x4 74x44 11 x. 2 Teach. Original inequality Add 4 to each side.. 17. Simplify.. 18. 19. 20. 21. 22. 23. Since 11 x is the same as x 11, the solution set is {xx 11}. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. The circle at 11 shows that 11 is not included in the inequality.. 13. 14. 15. 16. The heavy arrow pointing to the right shows that the inequality includes all numbers greater than 11.. SOLVE INEQUALITIES BY SUBTRACTION Subtraction can also be used to solve inequalities.. SOLVE INEQUALITIES BY SUBTRACTION. In-Class Example. Power Point®. 3 Solve q 23 14. Then graph the solution. {q | q 9} 12 11 10 9 8 7 6. Subtraction Property of Inequalities • Words. If any number is subtracted from each side of a true inequality, the resulting inequality is also true.. • Symbols. For all numbers a, b, and c, the following are true. 1. If a b, then a c b c. 2. If a b, then a c b c.. 17 8 17 5 8 5 12 3. • Example. This property is also true when and are replaced with. and .. Example 3 Solve by Subtracting Solve 19 r 16. Then graph the solution. 19 r 19 r 19 r. 16 16 19 3. Original inequality Subtract 19 from each side. Simplify.. Interactive. 3}.. The solution set is {rr. Chalkboard 8 7. 6. www.algebra1.com/extra_examples. 5. 4. 3. 2. 1. 0. 1. 2. 3. 4. 5. 6. 7. 8. PowerPoint® Presentations. Lesson 6-1 Solving Inequalities by Addition and Subtraction 319. Online Lesson Plans USA TODAY Education’s Online site offers resources and interactive features connected to each day’s newspaper. Experience TODAY, USA TODAY’s daily lesson plan, is available on the site and delivered daily to subscribers. This plan provides instruction for integrating USA TODAY graphics and key editorial features into your mathematics classroom. Log on to www.education.usatoday.com.. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools Lesson 6-1 Solving Inequalities by Addition and Subtraction 319.

(11) In-Class Examples. Terms with variables can also be subtracted from each side to solve inequalities.. Power Point®. Example 4 Variables on Both Sides. 4 Solve 12n 4 13n. Then graph the solution. {n | n 4} 6 5 4 3 2 1. Solve 5p 7 6p. Then graph the solution. 5p 7 6p 5p 7 5p 6p 5p 7p. 0. Teaching Tip. When students solve inequalities with variables on both sides, suggest that they subtract the term with the lesser coefficient from each side so the remaining coefficient of the variable will be positive.. Original inequality Subtract 5p from each side. Simplify.. Since 7 p is the same as p 7, the solution set is {pp 7}. 2 1. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Verbal problems containing phrases like greater than or less than can often be solved by using inequalities. The following chart shows some other phrases that indicate inequalities.. 5 Write an inequality for the sentence below. Then solve the inequality. Seven times a number is greater than six times that number minus two. 7a 6a 2; {a | a 2}. Inequalities . . • less than. • greater than. • fewer than. • more than. . . • at most. • at least. • no more than. • no less than. • less than or equal to. • greater than or equal to. 6 ENTERTAINMENT Alicia wants to buy season passes to two theme parks. If one season pass costs $54.99, and Alicia has $100 to spend on passes, the second season pass must cost no more than what amount? The second season pass must cost no more than $45.01.. Example 5 Write and Solve an Inequality Write an inequality for the sentence below. Then solve the inequality. Four times a number is no more than three times that number plus eight.. Yulia Barsukova of the Russian Federation won the gold medal in rhythmic gymnastics at the 2000 Summer Olympics in Sydney, and Yulia Raskina of Belarus won the silver medal. Source: www.olympic.org. 3n. eight.. . . 8. The solution set is {nn 8}.. Example 6 Write an Inequality to Solve a Problem OLYMPICS Yulia Raskina scored a total of 39.548 points in the four events of rhythmic gymnastics. Yulia Barsukova scored 9.883 in the rope competition, 9.900 in the hoop competition, and 9.916 in the ball competition. How many points did Barsukova need to score in the ribbon competition to surpass Raskina and win the gold medal?. Words. Barsukova’s total must be greater than Raskina’s total.. Variable. Let r Barsukova’s score in the ribbon competition.. . Barsukova’s total. Inequality 9.883 9.900 9.916 r 320. plus. . . . 4n. three times that number. is greater than. Raskina’s total.. . 39.548. . Solving Inequalities Ask students what the one phrase is that they will never see in a verbal inequality problem. Sample answer: equal, or is equal to. is no more than. Original inequality 4n 3n 8 4n 3n 3n 8 3n Subtract 3n from each side. n8 Simplify.. Olympics. Concept Check. Four times a number. Chapter 6 Solving Linear Inequalities. Differentiated Instruction. ELL. Verbal/Linguistic If students are having difficulty choosing the correct symbol for the problem’s wording, have them use the chart above to write the common inequality phrases on index cards and the appropriate inequality symbol on the back of each card. As students solve verbal problems such as Examples 5 and 6, they can pick the card that has the same wording as the problem. The back of the card will reveal the appropriate inequality symbol to use. 320. Chapter 6 Solving Linear Inequalities.

(12) Solve the inequality. 9.883 9.900 9.916 r 39.548 29.699 r 39.548 29.699 r 29.699 39.548 29.699 r 9.849. 3 Practice/Apply. Original inequality Simplify. Subtract 29.699 from each side. Simplify.. Study Notebook. Barsukova needed to score more than 9.849 points to win the gold medal.. Concept Check 1. Sample answers: y 1 2, y 1 4, y 3 0. Guided Practice GUIDED PRACTICE KEY Exercises. 1. OPEN ENDED List three inequalities that are equivalent to y 3. 2. Compare and contrast the graphs of a 4 and a 4. See margin.. 1 –4 5 6. 5–10. See pp. 365A– 365D.. than or equal to 5.. 4. Which graph represents the solution of m 3 7? a a.. b. 0. Examples. 4–10 11, 12 13. 5} means. The set of all numbers b such that b is greater. 3. Explain what {bb. 1. 2. 3. 4. 5. 6. 7. 0. 8. c.. 1. 2. 3. 4. 5. 6. 7. 8. d. 0. 1. 2. 3. 4. 5. 6. 7. 0. 8. 1. 2. 3. 4. 5. 6. 7. 8. Solve each inequality. Then check your solution, and graph it on a number line. 5. a 4 2 {aa 2} 8. y 2.5 3.1. {yy 5.6}. 6. 9 b 4 {bb 5} 9. 5.2r 6.7 6.2r. 7. t 7. 5 {tt 12}. 10. 7p 6p 2. {rr 6.7}. {pp 2}. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 6. • write a solution set in set-builder notation with an explanation of how to read it. • include an explanation of the difference between an inequality graph with a dot and one with a circle. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Define a variable, write an inequality, and solve each problem. Then check your solution. 11–12. Sample answer: Let n the number. 11. A number decreased by 8 is at most 14. n 8 14; {nn 22}. 12. A number plus 7 is greater than 2. n 7 2; {nn 5}. Application. 13. HEALTH Chapa’s doctor recommended that she limit her fat intake to no more than 60 grams per day. This morning, she ate two breakfast bars with 3 grams of fat each. For lunch she ate pizza with 21 grams of fat. If she follows her doctor’s advice, how many grams of fat can she have during the rest of the day?. no more than 33 g ★ indicates increased difficulty. Practice and Apply Homework Help For Exercises. See Examples. 14– 39 40– 45 46–55. 1– 4 5 6. Extra Practice. 2 d. 15. x 7 6 f 16. 4x 3x 1 a. a. 4 3. 2. 1. 0. 1. 2. 3. 4. 4 3. 2. 1. 0. 1. 2. 3. 4. b. c. 4 3. See page 833.. 17. 8 x 9 c 18. 5 x 6 e 19. x 1 0 b. 2. 1. 0. 1. 2. 3. 4. Assignment Guide. d. 4 3. 2. 1. 0. 1. 2. 3. 4. 4 3. 2. 1. 0. 1. 2. 3. 4. Basic: 15–31 odd, 41–53 odd, 56–76 Average: 15–53 odd, 56–76 Advanced: 14–52 even, 53-68 (optional: 69–76). e. f. 4 3. www.algebra1.com/self_check_quiz. Organization by Objective • Solve Inequalities by Addition: 14–51 • Solve Inequalities by Subtraction: 14–51 Odd/Even Assignments Exercises 14–51 are structured so that students practice the same concepts whether they are assigned odd or even problems.. Match each inequality with its corresponding graph. 14. x 3. About the Exercises…. 2. 1. 0. 1. 2. 3. 4. Lesson 6-1 Solving Inequalities by Addition and Subtraction 321. Answer. Unlocking Misconceptions Rewriting Inequalities An equation such as x 5 can be rewritten as 5 x because of the Symmetric Property of Equality. Because of this property, students may incorrectly assume that they can rewrite an inequality such as 3 y as y 3. Remind students that the inequality sign always points to the smaller value. In 3 y, it points to y, so to write the expression with y on the left, use y 3.. 2. In both graphs, the line is darkened to the left. In the graph of a 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a 4, there is a dot at 4 to indicate that 4 is included in the graph.. Lesson 6-1 Solving Inequalities by Addition and Subtraction 321.

(13) NAME ______________________________________________ DATE. ____________ PERIOD _____. Solve each inequality. Then check your solution, and graph it on a number line.. 20–37. See pp. 365A–365D.. Study Guide andIntervention Intervention, 6-1 Study Guide and. p. 343 p.and 344 Solving(shown) Inequalities byand Addition Subtraction. 20. t 14. Addition can be used to solve inequalities. If any number is added to each side of a true inequality, the resulting inequality is also true. For all numbers a, b, and c, if a b, then a c b c, and if a b, then a c b c.. The property is also true when and are replaced with. Example 1. and .. Example 2. Solve x 8 6. Then graph it on a number line.. Solve 4 2a a. Then graph it on a number line.. x 8 6 x 8 8 6 8 x2. 4 2a a 4 2a 2a a 2a 4a a4. Original inequality Add 8 to each side. Simplify.. The solution in set-builder notation is {x|x 2}. Number line graph: 4 3 2 1. 0. 1. 2. 3. Original inequality Add 2a to each side. Simplify. 4 a is the same as a 4.. The solution in set-builder notation is {a|a 4}. Number line graph: 2 1. 4. 0. 1. 2. 3. 4. 5. 6. Exercises Solve each inequality. Then check your solution, and graph it on a number line. 16 {tt 28}. 1. t 12. 2. n 12 6 {nn 18}. 26 27 28 29 30 31 32 33 34. 3. 6 g 3 {gg 9}. 12 13 14 15 16 17 18 19 20. 7. 8. 9 10 11 12 13 14 15. 4. n 8 13 {nn 5} 5. 12 12 y {yy 0} 6. 6 s 8 {ss 2} 10 9 8 7 6 5 4 3 2. 4 3 2 1 0. 1. 2. 3. 4 3 2 1 0. 4. 1. 2. 3. Lesson 6-1. Addition Property of Inequalities. 28. {yy 8} 31. {ww 1} 32. {vv 1} 33. {aa 5} 34. {hh 0.36} 35. {xx 0.6} 36. aa. 37. pp. . 1 8 1 1 9. 12 0.4n. 8. 0.6n. {xx 8}. 11. z . ★ 33. a (2) 3. ★ 34. 0.23 h (0.13). ★ 36. a 14 18. ★ 37. p 23 49. a. d. n 3 12; {nn 15}. 15. Forty is no greater than the difference of a number and 2. 40 n 2; {nn 42} ____________ PERIOD _____. p. 345 and Practice, p. 346by(shown) Solving Inequalities Addition and Subtraction. Match each inequality with its corresponding graph. x 15 b. a. 6 5 4 3 2 1 0. b.. 3. 8x 7x 4 a. c.. 4. 12 x 9 c. d.. 0. 1. 2. 3. 4. 5. 6. 1. 2. 7. 8. 1. 2. 3. 4. 5. 6. 7. ?. 20 8. ★ 39. If z 2 10, then complete each inequality. a. z ? 12 b. z ? 5 7. c. d 5. ?. 7. c. z 4 . ?. 16. 41. A number decreased by 5 is less than 33. n 5 33; {nn 38} 42. Thirty is no greater than the sum of a number and 8.. 42. 30 n (8); {nn 38} 43. 2n n 14; {nn 14} 44. n ( 7) 18; {nn 25}. 43. Twice a number is more than the sum of that number and 14. 44. The sum of two numbers is at most 18, and one of the numbers is 7. 45. Four times a number is less than or equal to the sum of three times the number and 2. 4n 3n (2); {nn 2} 46. BIOLOGY Adult Nile crocodiles weigh up to 2200 pounds. If a young Nile crocodile weighs 157 pounds, how many pounds might it be expected to gain in its lifetime? no more than 2043 lb. 8 7 6 5 4 3 2 1 0 0. b. d . 40. The sum of a number and 13 is at least 27. n 13 27; {nn 14}. 14. The difference of two numbers is more than 12, and one of the numbers is 3.. 2. 4x 3 5x d. 12. Define a variable, write an inequality, and solve each problem. Then check your solution. 40–45. Sample answer: Let n the number.. 13. A number decreased by 4 is less than 14. n 4 14; {nn 18}. 1. 8. 17, then complete each inequality. ?. 4. Define a variable, write an inequality, and solve each problem. Then check your solution. 13–15. Sample answer: Let n the number.. NAME ______________________________________________ DATE. 2.3. ★ 38. If d 5. {bb 4}. Skills Practice, 6-1 Practice (Average). 25. 4 8 r {rr 4}. ★ 35. x 1.7. 12. 2b 4 3b. zz 1 23 . {yy 25}. 22. n 7 3 {nn 4}. ★ 32. v (4) 3. {kk 12}. 4 3. 1 3. 10. y 10 15 2y. 24. 5 3 g {gg 2}. q 7 {qq 4} 27. 2 m 1 {mm 3} 28. 2y 8 y 29. 3f 3 2f {ff 3} 30. 3b 2b 5 {bb 5}31. 4w 3w 1. 9. 8k 12 9k. {nn 12}. 21. d 5 7 {dd 2}. 26. 3. Solve each inequality. Then check your solution. 7. 3x 8 4x. 18 {tt 4}. 23. s 5 1 {ss 4}. Solve Inequalities by Addition. 8. Solve each inequality. Then check your solution, and graph it on a number line. 5. r (5) 2 {rr 7} 8 7 6 5 4 3 2 1 0. 2. 5 {nn 2.5}. 7. n 2.5. 4 3 2 1 0. 2 3. 1. 2. 3. . 4 3 2 1 0. 1. 2. 3. 4. 5. 6. 4 3 2 1 0. 1 3. 7. 8. 9 10. 8. 1.5 y 1 {yy 0.5}. 4. 9. z 3 zz 2. 3. 47. ASTRONOMY There are at least 200 billion stars in the Milky Way. If 1100 of these stars can be seen in a rural area without the aid of a telescope, how many stars in the galaxy cannot be seen in this way? at least 199,999,998,900 stars. 4x {xx 8}. 6. 3x 8. . 1 2. 3 4. 10. c . 4 3 2 1 0. 4. 1. 2. 3. 4. cc 1 14 1. 2. 3. 48. BIOLOGY There are 3500 species of bees and more than 600,000 species of insects. How many species of insects are not bees? more than 596,500 species. 4. Define a variable, write an inequality, and solve each problem. Then check your solution. 11–14. Sample answer: Let n the number. 11. The sum of a number and 17 is no less than 26.. 49. BANKING City Bank requires a minimum balance of $1500 to maintain free checking services. If Mr. Hayashi knows he must write checks for $1300 and $947, how much money should he have in his account before writing the checks? at least $3747. n 17 26; {nn 9}. 12. Twice a number minus 4 is less than three times the number.. 2n 4 3n; {nn 4}. 13. Twelve is at most a number decreased by 7.. 12 n 7; {nn 19}. 14. Eight plus four times a number is greater than five times the number.. 8 4n 5n; {nn 8}. 15. ATMOSPHERIC SCIENCE The troposphere extends from the earth’s surface to a height of 6–12 miles, depending on the location and the season. If a plane is flying at an altitude of 5.8 miles, and the troposphere is 8.6 miles deep in that area, how much higher can the plane go without leaving the troposphere? no more than 2.8 mi 16. EARTH SCIENCE Mature soil is composed of three layers, the uppermost being topsoil. Jamal is planting a bush that needs a hole 18 centimeters deep for the roots. The instructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants to add even more cushion, and the topsoil in his yard is 30 centimeters deep, how much more cushion can he add and still remain in the topsoil layer? no more than 4 cm NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 6-1 Readingto to Learn Learn MathematicsELL Mathematics, p. 347 Solving Inequalities by Addition and Subtraction Pre-Activity. How are inequalities used to describe school sports? Read the introduction to Lesson 6-1 at the top of page 318 in your textbook. • Use the information in the graph to write an inequality statement about participation in two sports. Sample answer:. 50. GEOMETRY The length of the base of the triangle at the right is less than the height of the triangle. What are the possible values of x? more than 8 in.. Biology One common species of bees is the honeybee. A honeybee colony may have 60,000 to 80,000 bees.. 12 in.. 51. SHOPPING Terrell has $65 to spend at the mall. He bought a T-shirt for $18 and a belt for $14. If Terrell still wants to buy a pair of jeans, how much can he spend on the jeans? no more than $33. Source: Penn State, Cooperative Extension Service. For softball and track and field, 13,009 14,587. ★ 52. SOCCER The Centerville High School soccer team plays 18 games in the. • Rewrite your inequality statement to show that 40 schools added both of the sports. Is the statement still true?. season. The team has a goal of winning at least 60% of its games. After the first three weeks of the season, the team has won 4 games. How many more games must the team win to meet their goal? at least 7 more games. Sample answer: 13,049 14,627; yes. Reading the Lesson Write the letter of the graph that matches each inequality. 1. x 1. b. 1. d. b.. 3. x 1. a. c.. 2. x. 4. x 1. c. 322. Chapter 6 Solving Linear Inequalities. a. 3 2 1 0. 1. 2. 3. 3 2 1 0. 1. 2. 3. 3 2 1 0. 1. 2. 3. 3 2 1 0. 1. 2. 3. NAME ______________________________________________ DATE. 6-1 Enrichment Enrichment,. d.. 5. Use the chart to write a sentence that could be described by the inequality 3n Then solve the inequality.. . . . . less than fewer than. greater than more than. at most no more than less than or equal to. at least no less than greater than or equal to. Sample answer: Three times a number is at least two times the number plus 7; n 7. Helping You Remember 6. Teaching someone else can help you remember something. Explain how you would teach another student who missed class to solve the inequality 2x 4 3x.. Triangle Inequalities Recall that a line segment can be named by the letters of its endpoints. Line segment AB (written as AB ) has points A and B for endpoints. The length of AB is written without the bar as AB. AB BC. mA mB. The statement on the left above shows that AB is shorter than B C . The statement on the right above shows that the measure of angle A is less than that of angle B. These three inequalities are true for any triangle ABC, no matter how long the sides. a. AB BC AC b. If AB AC, then mC mB. c. If mC mB, then AB AC.. Subtract 2x from each side. Simplify. Use the three triangle inequalities for these problems. 1 List the sides of triangle DEF in order of increasing length. 322. ____________ PERIOD _____. p. 348. 2n 7.. Inequalities. Chapter 6 Solving Linear Inequalities. 4 x in.. A. B. C.

(14) 53. CRITICAL THINKING Determine whether each statement is always, sometimes, or never true. a. If a b and c d, then a c b d. always b. If a b and c d, then a c. b d. never. Open-Ended Assessment. c. If a b and c d, then a c b d. sometimes HEALTH For Exercises 54 and 55, use the following information. Hector’s doctor told him that his cholesterol level should be below 200. Hector’s cholesterol is 225. 54. Let p represent the number of points Hector should lower his cholesterol. Write an inequality with 225 p on one side. 225 p 200. ★ 55. Solve the inequality. {pp 25} 56. WRITING IN MATH. Answer the question that was posed at the beginning of the lesson. See pp. 365A–365D.. How are inequalities used to describe school sports? Include the following in your answer: • an inequality describing the number of schools needed to add girls’ track and field so that the number is greater than the number of schools currently participating in girls’ basketball.. Standardized Test Practice. A. A B C D. x75. B. x 4 16. C. x 1 13. D. 12. x. 5? A. The sum of a number and six is at least five. The sum of a number and six is at most five. The sum of a number and six is greater than five. The sum of a number and six is no greater than five.. 59. Would a scatter plot for the relationship of a person’s height to the person’s grade on the last math test show a positive, negative, or no correlation? (Lesson 5-7). no Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of each equation. (Lesson 5-6) 60. (1, 3); y 3x 2. Getting Ready for the Next Lesson. 61. (0, 4); x y 3. y 3x 6. 62. (1, 2); 2x y 1. y x 4. y 2x 4. Find the next two terms in each sequence. (Lesson 4-8) 63. 7, 13, 19, 25, …. 64. 243, 81, 27, 9, …. 31, 37. 65. 3, 6, 12, 24, …. 3, 1. 48, 96. Solve each equation if the domain is {1, 3, 5}. (Lesson 4-4) 66. y 2x. 67. y 7 x. 68. 2x y 6. PREREQUISITE SKILL Solve each equation. (For review of multiplication and division equations, see Lesson 3-3.). 69. 6g 42 7. t 70. 14 126. 4 73. x 28 7. 74. 5.3g 11.13 2.1. 49. 9. 2 71. y 14 21 3 a 75. 7 3.5. Intervention Students should always check their solutions, but they often hurry to finish their assignments and omit this step. Remind students that checking solutions is especially important with inequalities because the direction of the inequality sign often gets changed when writing solutions in set-builder notation.. New. Maintain Your Skills. 66. {(1, 2), (3, 6), (5, 10)} 67. {(1, 8), (3, 4), (5, 2)} 68. {(1, 8), (3, 0), (5, 4)}. Modeling Draw a large number line on an overhead transparency. Using a washer or a coin, and a ribbon or a strip of paper, graph several different inequalities on the overhead. Use the coin to indicate a dot on the number line and the washer to indicate a circle. The ribbon is the ray representing points greater or less than the starting value. Have students identify the inequalities you graph, identifying the significance of the circle or dot.. 57. Which inequality is not equivalent to x 12? C 58. Which statement is modeled by n 6. Mixed Review. 4 Assess. 24.5. 72. 3m 435 145 76. 8p 35 4.375. Lesson 6-1 Solving Inequalities by Addition and Subtraction 323. Getting Ready for Lesson 6-2 PREREQUISITE SKILL In Lesson 6-2, students learn how to solve inequalities using multiplication and division. The process is almost identical to the process for solving equations using multiplication and division. Use Exercises 69–76 to determine your students’ familiarity with solving these types of equations.. Lesson 6-1 Solving Inequalities by Addition and Subtraction 323.

(15) Algebra Activity. A Preview of Lesson 6-2. A Preview of Lesson 6-2. Solving Inequalities. Getting Started. You can use algebra tiles to solve inequalities.. Objective Use algebra tiles to solve inequalities.. Solve 2x 6.. Materials algebra tiles equation mat self-adhesive notes. Model the inequality.. Remove zero pairs.. Use a self-adhesive note to cover the equals sign on the equation mat. Then write a symbol on the note. Model the inequality.. Since you do not want to solve for a negative x tile, eliminate the negative x tiles by adding 2 positive x tiles to each side. Remove the zero pairs.. Teach . x. 1. 1. 1. 1. 1. 1. 1. x. 1. 1. 1. 1. x. x. x. x 2x 2x 6 2x. 2x 6. Remove zero pairs.. Group the tiles.. Add 6 negative 1 tiles to each side to isolate the x tiles. Remove the zero pairs.. Assess Have students work in small groups for Exercises 1–7. Observe to determine if they are able to verbalize the activities in Exercises 1–4. Students should conclude after Exercises 6–7 that when they multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.. 1. x. • You may wish to do the example as a demonstration. • Make sure the inequality sign on the self-adhesive note is pointed in the correct direction to match the inequality. • Once they have isolated the x tiles, remind students to separate the 1 tiles into equal groups to correspond to the number of x tiles. • If the x tiles end up on the right side of the inequality, students may rotate the mat 180 degrees to read the inequality with the variable on the left side.. x. . 1 1. 1 1. 1 1. 1 1. 1 1. 1 1. 1. 1. 1. 1. x. 1. 1. x. Separate the tiles into 2 groups.. 1. 1. 1. x. 1. 1. 1. x. 3 x or x 3. 6 2x. Model and Analyze Use algebra tiles to solve each inequality. 1. 4x 12 {xx 3} 2. 2x 8 {xx 4} 3. 3x 6 {xx 2} 4. 5x 5 {xx 1} 5. In Exercises 1–4, is the coefficient of x in each inequality positive or negative? negative 6. Compare the inequality symbols and locations of the variable in Exercises 1–4 with those in their solutions. What do you find? 6–7. See pp. 365A–365D. 7. Model the solution for 2x 6. What do you find? How is this different from solving 2x 6?. 324. Chapter 6 Solving Linear Inequalities. Resource Manager. Study Notebook You may wish to have students summarize this activity and what they learned from it.. 324. Chapter 6 Solving Linear Inequalities. Teaching Algebra with Manipulatives. Glencoe Mathematics Classroom Manipulative Kit. • pp. 10–11 (master for algebra tiles) • p. 16 (master for equation mat) • p. 115 (student recording sheet). • algebra tiles • equation mat.

(16) Solving Inequalities by Multiplication and Division • Solve linear inequalities by using multiplication.. 1 Focus. • Solve linear inequalities by using division.. are inequalities important in landscaping? Isabel Franco is a landscape architect. To beautify a garden, she plans to build a decorative wall of either bricks or blocks. Each brick is 3 inches high, and each block is 12 inches high. Notice that 3 12.. 48 in.. 12 in.. A wall 4 bricks high would be lower than a wall 4 blocks high. 4. ?. 12. 4. 12 48. SOLVE INEQUALITIES BY MULTIPLICATION If each side of an inequality is multiplied by a positive number, the inequality remains true. 8 5 8(2). 5 9 5(4). 5(2) Multiply each side by 2.. ?. 16 10. ?. 9(4) Multiply each side by 4.. 20 36. This is not true when multiplying by negative numbers. 5 3 5(2). ?. 6 8. 3(2) Multiply each side by 2.. 10 6. 6(5). 5-Minute Check Transparency 6-2 Use as a quiz or a review of Lesson 6-1. Mathematical Background notes are available for this lesson on p. 316C.. Building on Prior Knowledge. 12 in.. 3 in.. 3. Lesson Notes. ?. 8(5) Multiply each side by 5.. 30 40. If each side of an inequality is multiplied by a negative number, the direction of the inequality symbol changes. These examples illustrate the Multiplication Property of Inequalities .. Multiplying by a Positive Number • Words. If each side of a true inequality is multiplied by the same positive number, the resulting inequality is also true.. • Symbols. If a and b are any numbers and c is a positive number, the following are true. If a b, then ac bc, and if a b, then ac bc.. The process of solving inequalities is identical to the process of solving equations except when multiplying or dividing by a negative value. Students should understand that they do not have to learn a whole new process, but just a special rule when using negatives. are inequalities important in landscaping? Ask students: • What is similar about the two walls? What is different? Both walls are 4 rows high. Each row of bricks is 3 inches high and each row of blocks is 12 inches high. • By what number are both sides of the inequality 3 12 multiplied to yield 12 48? 4 • After you multiply both sides of the inequality 3 12 by the same number to yield 12 48, is the inequality still true? Explain. Yes; 12 is less than 48.. Lesson 6-2 Solving Inequalities by Multiplication and Division 325. Resource Manager Workbook and Reproducible Masters Chapter 6 Resource Masters • Study Guide and Intervention, pp. 349–350 • Skills Practice, p. 351 • Practice, p. 352 • Reading to Learn Mathematics, p. 353 • Enrichment, p. 354 • Assessment, p. 393. Parent and Student Study Guide Workbook, p. 47 School-to-Career Masters, p. 12. Transparencies 5-Minute Check Transparency 6-2 Answer Key Transparencies. Technology Interactive Chalkboard. Lesson x-x Lesson Title 325.

(17) Multiplying by a Negative Number. 2 Teach SOLVE INEQUALITIES BY MULTIPLICATION. In-Class Examples Teaching Tip. If a and b are any numbers and c is a negative number, the following are true. If a b, then ac bc, and if a b, then ac bc. and .. You can use this property to solve inequalities.. 7. Example 1 Multiply by a Positive Number. isolate b, multiply by the. b 7. 1 reciprocal of , which is 7.. Solve 25. Then check your solution.. 7. b 7 b (7) 7. g. 1 Solve 12. Then check 3 your solution. {g | g 36} 2. • Symbols. This property also holds for inequalities involving. 1 b that is the same as b. To. 3 Solve d 4. If each side of a true inequality is multiplied by the same negative number, the direction of the inequality symbol must be reversed so that the resulting inequality is also true.. Power Point®. Remind students. 7. • Words. b. 6. {d | d 8}. 25. Original inequality. (7)25. Multiply each side by 7. Since we multiplied by a positive number, the inequality symbol stays the same.. 175. CHECK To check this solution, substitute 175, a number less than 175, and a number greater than 175 into the inequality.. 3 Write an inequality for the sentence below. Then solve the inequality. Four-fifths of a number is at most twenty.. Let b 175.. Let b 140.. Let b 210.. 175 ? 25 7. 140 ? 25 7. 210 ? 25 7. 25 ⻫. 25. 4 r 20; {r | r 25} 5. 20. The solution set is {bb. Study Tip Common Misconception A negative sign in an inequality does not necessarily mean that the direction of the inequality should change. For example, when solving x 3, do not change 6. the direction of the inequality.. 25. 30. 25 ⻫. 175}.. Example 2 Multiply by a Negative Number 2 5 2 p 14 5 5 2 5 p (14) 2 5 2. Solve p 14.. . . . . p 35. Original inequality 5 2. Multiply each side by and change to .. The solution set is {pp 35}.. Example 3 Write and Solve an Inequality Write an inequality for the sentence below. Then solve the inequality. One fourth of a number is less than 7.. 1 n 7 4 1 (4)n (4)(7) 4. n 28 326. 326. Chapter 6 Solving Linear Inequalities. Chapter 6 Solving Linear Inequalities. a number. . is less than. . 1 4. 7.. . of. . . One fourth. n. . 7. Original inequality Multiply each side by 4 and do not change the inequality’s direction.. The solution set is {nn 28}..

(18) SOLVE INEQUALITIES BY DIVISION Dividing each side of an inequality by the same number is similar to multiplying each side of an equality by the same number. Consider the inequality 6 15. Divide each side by 3.. Divide each side by 3.. 6. 6. . 15. 3. ?. 15. 2. . 5. SOLVE INEQUALITIES BY DIVISION. 3. 6. Since each side is divided by a positive number, the direction of the inequality symbol remains the same.. 6. . 15. (3). ?. 15. 2. . 5. (3). Assessment Challenge students to explain how they might already know how to solve inequalities by division. Students should suggest that since they know how to solve inequalities by multiplication, and since division is the same as multiplying by a reciprocal, then they already know how to solve inequalities by division.. New. Since each side is divided by a negative number, the direction of the inequality symbol is reversed.. These examples illustrate the Division Property of Inequalities.. Dividing by a Positive Number • Words. If each side of a true inequality is divided by the same positive number, the resulting inequality is also true.. • Symbols. If a and b are any numbers and c is a positive number, the following are true. a b a b If a b, then , and if a b, then . c. c. c. c. Dividing by a Negative Number • Words. If each side of a true inequality is divided by the same negative number, the direction of the inequality symbol must be reversed so that the resulting inequality is also true.. • Symbols. If a and b are any numbers and c is a negative number, the following are true. a b a b If a b, then , and if a b, then . c. c. c. c. and .. This property also holds for inequalities involving. Example 4 Divide by a Positive Number Original inequality. 14h 91 14 14. Divide each side by 14 and do not change the direction of the inequality sign.. Power Point®. Teaching Tip. Point out to students that the rules for the Division Property of Inequalities state that each side of an inequality can be divided by a positive or negative number. In neither case is zero included because division by zero is an undefined operation.. 4 Solve 12s. Solve 14h 91. 14h 91. In-Class Example. 60. {s | s 5}. h 6.5 CHECK Let h 6.5.. Let h 7.. Let h 6.. 14h 91. 14h 91. 14h 91. ?. ?. 14(6.5) 91 91. 14(7) 91 98 91 ⻫. 91. ?. 14(6) 91 84. 91. The solution set is {hh 6.5}.. Since dividing is the same as multiplying by the reciprocal, there are two methods to solve an inequality that involve multiplication.. www.algebra1.com/extra_examples. Lesson 6-2 Solving Inequalities by Multiplication and Division 327. Differentiated Instruction Kinesthetic Have students write an inequality involving a negative coefficient of the variable on their paper, using a self-adhesive note for the inequality symbol, such as 12x 24. Tell them that they are going to change all the signs in the inequality, so everything is its opposite. The expression becomes 12x 24. Students now can divide without having to worry about the inequality sign.. Lesson 6-2 Solving Inequalities by Multiplication and Division 327.

(19) In-Class Examples. Example 5 Divide by a Negative Number. Power Point®. Solve 5t 275 using two methods.. Teaching Tip. Point out to students that it may be easier to solve an inequality using division when the inequality involves whole numbers, and easier to solve using multiplication by reciprocals when the inequality involves fractions.. Method 1 Divide. 5t. Divide each side by 5 and change. Method 2. to .. Multiply by the multiplicative inverse.. 5t. . 275. . . Original inequality. . 1 1 (5t) 275 5 5. t 55. Teaching Tip. Another way to check the solution is to rework the problem using a different method.. A 2x 12 B 6x 72. Original inequality. t 55 Simplify.. 5 Solve 8q 136 using two methods. {q |q 17}. 6 Which inequality does not have the solution {x | x 6}? B. 275. 5t 275 5 5. 1 5. Multiply each side by and change. to .. Simplify.. The solution set is {tt 55}.. You can use the Multiplication Property and the Division Property for Inequalities to solve standardized test questions.. Standardized Example 6 The Word “not” Test Practice Multiple-Choice Test Item. 5 6. C x 5. Which inequality does not have the solution {yy 5}? A. 1 3 D x 8 4. 7y. B. 35. 2y 10. C. 7 y 5. 7. D. y 4. . 5 4. Read the Test Item You want to find the inequality that does not have the solution set {yy 5}.. 3 Practice/Apply. Solve the Test Item Consider each possible choice. A. Test-Taking Tip. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 6. • copy the rules for the Multiplication Property of Inequalities and Division Property of Inequalities. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Concept Check. y 5. C. B. 35. 7y 35 7 7. Always look for the word not in the questions. This indicates that you are looking for the one incorrect answer, rather than looking for the one correct answer. The word not is usually in italics or uppercase letters to draw your attention to it.. 2. Sample answer: Three fourths of a number is greater than 9.. 7y. 7 y 5 5 7 y 7 5. . y. y 5. ⻫. 7. 57(7) 5. 2y 10 2y 10 2 2. D. ⻫. y 5 4 4 y 5 (4) (4) 4 4. . . . y 5. The answer is C.. 1. Explain why you can use either the Multiplication Property of Inequalities or the Division Property of Inequalities to solve 7r 28. See margin.. Answer multiplying each side by . 1 or 7. by dividing each side by 7. In either case, you must reverse the inequality.. 328. Chapter 6 Solving Linear Inequalities. 3 4. 2. OPEN ENDED Write a problem that can be represented by the inequality c 9.. 328 Chapter 6 Solving Linear Inequalities. 1. You could solve the inequality by. ⻫. Example 6 Some questions can be answered without solving each equation or inequality given. Have students examine each inequality in Example 6 to determine what inequality sign should be included in the solution set without working it out. For A and D, the sign becomes . For B, it stays . C is the remaining choice.. Standardized Test Practice.

(20) 3. Ilonia; when you divide each side of an inequality by a negative number, you must reverse the inequality sign.. 3. FIND THE ERROR Ilonia and Zachary are solving 9b 18.. Ilonia. Zachary. –9b ≤ 18. –9b ≤ 18. 18 –9b ≥ –9 –9. –9b 18 ≤ –9 –9. b ≥ –2. b ≤ –2. FIND THE ERROR Tell students to look first at the solutions from Ilonia and Zachary. The only difference is the inequality symbol. Since the solution involves division by a negative number, the inequality symbol of the solution must be reversed from the original inequality.. Who is correct? Explain your reasoning.. Guided Practice. a. Seven times a number is at least 14.. GUIDED PRACTICE KEY Exercises. Examples. 4, 5, 10, 11 6–9 12. 3 1, 2, 4, 5 6. 14? a. 4. Which statement is represented by 7n. b. Seven times a number is greater than 14. c. Seven times a number is at most 14. d. Seven times a number is less than 14.. About the Exercises…. 5. Which inequality represents five times a number is less than 25? c a. 5n 25. b. 5n. 25. c. 5n 25. Organization by Objective • Solve Inequalities by Multiplication: 13–50 • Solve Inequalities by Division: 13–50. d. 5n 25. Solve each inequality. Then check your solution. t 7. 12. 6. 15g 75. 9. {gg 5}. {tt 108}. 2 3. 8. b 9. 9. 25f. {bb 13.5}. 9. {ff 0.36}. Define a variable, write an inequality, and solve each problem. Then check your solution. 10–11. Sample answer: Let n the number.. Odd/Even Assignments Exercises 13–50 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 10. The opposite of four times a number is more than 12. 4n 12; {nn 3} 1 11. Half of a number is at least 26. n 26; {nn 52} 2. Standardized Test Practice. 12. Which inequality does not have the solution {xx 4}? B A. 5x 20. B. 6x 24. C. 1 4 x 5 5. D. 3 4. x 3. Assignment Guide. ★ indicates increased difficulty. Basic: 13–29 odd, 35, 39, 41, 45, 47, 49, 52, 55–78 Average: 13–51 odd, 52, 53, 55–78 Advanced: 14–54 even, 55–72 (optional: 73–78) All: Practice Quiz 1 (1–10). Practice and Apply Homework Help. Match each inequality with its corresponding statement.. For Exercises. See Examples. 1 13. n 10 d. a. Five times a number is less than or equal to ten.. 13–18, 39–44 19–38 45–51. 3. 14. 5n 10 a. b. One fifth of a number is no less than ten.. 1, 2, 4, 5 6. 15. 5n 10 e. c. Five times a number is less than ten.. 16. 5n 10 f. d. One fifth of a number is greater than ten.. 1 17. n 5. e. Five times a number is greater than ten.. Extra Practice See page 833.. 5. 10 b. 18. 5n 10 c. f. Negative five times a number is less than ten.. Solve each inequality. Then check your solution. 19–34. See margin. 19. 6g 144. 20. 7t 84. m 23. 5 5 27. y 8. b 24. 5 10 2 28. v 6 3. 7 15. ★ 31. 2.5w 6.8 www.algebra1.com/self_check_quiz. ★ 32. 0.8s 6.4. 21. 14d. 84. r 25. 7 7 3 29. q 33 4 3 15c ★ 33. 14 7. 22. 16z 64 a 11 2 30. p 10 5 4m 3 ★ 34. 5 15. 26. 9. Lesson 6-2 Solving Inequalities by Multiplication and Division 329. Answers 19. {g | g 24} 20. {t |t 12} 21. {d |d 6} 22. {z | z 4} 23. {m |m 35} 24. {b |b 50}. 25. {r |r 49} 26. {a |a 99} 27. {y |y 24} 28. {v |v 9} 29. {q |q 44} 30. {p | p 25}. 31. {w |w 2.72} 32. {s |s 8}. 35.. 1 10 1 34. m | m 4. 36.. 33. c |c . 6 5 4 3 2 1. 0. 1. 2. 3. 4. 5. 0. 6. 1. 7. 2. 8. Lesson 6-2 Solving Inequalities by Multiplication and Division 329.

(21) NAME ______________________________________________ DATE. y 1 8 2 m 1 36. Solve . Then graph the solution. 9 3. 35. Solve . Then graph the solution. {yy 4}; See margin for graph.. ____________ PERIOD _____. Study Guide andIntervention Intervention, 6-2 Study Guide and. p. 349 p. 350 Solving(shown) Inequalities byand Multiplication and Division. Solve Inequalities by Multiplication If each side of an inequality is multiplied by the same positive number, the resulting inequality is also true. However, if each side of an inequality is multiplied by the same negative number, the direction of the inequality must be reversed for the resulting inequality to be true.. ★ 37. If 2a. 1. if c is positive and a b, then ac bc; if c is positive and a b, then ac bc; 2. if c is negative and a b, then ac bc; if c is negative and a b, then ac bc.. Example 1 y 8. Solve . 12. y 96. Multiply each side by 8; change. to .. Original equation. 34 34 k 43 15. 4 Multiply each side by 3.. k 20. Simplify.. 3 k 15. 4. Solve. 3 k 15 4. Original equation. 8y . (8) (8)12. and .. Example 2. y 12. 8. Simplify.. The solution is {kk 20}.. The solution is {yy 96}.. n 50. {yy 12} 1 4. 5. n. {nn 1100} 2 3. 1 3. 3 4. 2. 9p 5. . {gg 10}. 4. 6. {hh 5} 3m 5. {pp 36}. 3 20. 2h 4. 8. 2.51 . mm 14 n 10. 10. . pp 5 12. p 6. 3. 7. . bb 21 . {nn 40} g 5. 3. h. 6. b . 10. 9. . 3 5. 2. 22. 11. . . {hh 5.02} 2a 7. 12. . 5.4. {nn 54}. 6. {aa 21}. 43. 0.25n 90; {nn 360}. 1 n 14; {nn 28} 2 1 3 1 15. One fifth of a number is at most 30. n 30; {nn 150} 5. 14. The opposite of one-third a number is greater than 9. n 9; {nn 27}. ____________ PERIOD _____. p. 351 and Practice, p. 352by(shown) Solving Inequalities Multiplication and Division. Match each inequality with its corresponding statement. 1. 4n. 5 d. More About . . .. a. Negative four times a number is less than five.. 4 5. 2. n 5 f. b. Four fifths of a number is no more than five.. 3. 4n 5 e. c. Four times a number is fewer than five.. 4 5. 4. n 5 b. d. Negative four times a number is no less than five.. 5. 4n 5 c. e. Four times a number is at most five.. 6. 4n 5 a. f. Four fifths of a number is more than five.. 8. 13h 52. {aa 70}. s 9. 16. {hh 4}. 2 3. 5 9. 11. n 12. 12. t 25. {nn 18}. 16. 0.9c 9. {bb 0.25} 19. 15y 3. 3 5. 13. m 6. {tt 45}. 15. 3b 0.75. yy 51 . 20.8. 10. 39 13p. {pp 3} 10 3. 14. k. {mm 10} 17. 0.1x. {cc 10} 20. 2.6v. 6. {ss 96}. 4. 21. 0 0.5u. {vv 8}. {jj 9.2} 7 8. 22. f 1. ff 87 . Define a variable, write an inequality, and solve each problem. Then check your solution. 2325. Sample answer: Let n the number. 23. Negative three times a number is at least 57. 3n 57; {nn 19} 24. Two thirds of a number is no more than 10.. at least 139 people 49. LANDSCAPING Matthew is planning a circular flower garden with a low fence around the border. If he can use up to 38 feet of fence, what radius can he use for the garden? (Hint: C 2r) up to about 6 ft. j 18. 2.3 4. {uu 0}. 2 n 10; {nn 15} 3 3 n 6; {nn 10} 5. 25. Negative three fifths of a number is less than 6. . 26. FLOODING A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will exceed flood stage. How long can the river rise at this rate without exceeding flood stage? no more than 8 h. Zoos Dr. Harry Wegeforth founded the San Diego Zoo in 1916 with just 50 animals. Today, the zoo has over 3800 animals. Source: www.sandiegozoo.org. 27. SALES Pet Supplies makes a profit of $5.50 per bag on its line of natural dog food. If the store wants to make a profit of no less than $5225, how many bags of dog food does it need to sell? at least 950 bags NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 6-2 Readingto to Learn Learn Mathematics. ELL Mathematics, 353 Solving Inequalitiesp. by Multiplication and Division. Pre-Activity. Why are inequalities important in landscaping? Read the introduction to Lesson 6-2 at the top of page 325 in your textbook. • Would a wall 6 bricks high be lower than a wall 6 blocks high? Why?. yes; 6(3) 6(12). • Would a wall n bricks high be lower than a wall n blocks high? Explain.. yes; When one quantity is less than another quantity, multiplying both quantities by the same positive number does not change the truth of the inequality.. 1. Write an inequality that describes each situation.. 85. 330. b. Twelve times a number k is at least 7. 12k 7 c. A number x divided by 10 is less than or equal to 50. x d. Three fifths of a number n is at most 13.. a. If a b, then a2 b2.. 14 A number t divided by 3 is greater than or equal to 14.. c. 11x 32 11 times a number x is at most 32.. Helping You Remember 3. In your own words, write a rule for multiplying and dividing inequalities by positive and negative numbers.. Sample answer: When you multiply or divide each side of a true inequality by a positive number, the result is true. When you multiply or divide a true inequality by a negative number, you must reverse the direction of the inequality sign.. Chapter 6 Solving Linear Inequalities. 6-2 Enrichment Enrichment,. ____________ PERIOD _____. p. 354. The Maya 'I'he Maya were a Native American people who lived from about 1500 B.C. to about 1500 A.D. in the region that today encompasses much of Central America and southern Mexico. Their many accomplishments include exceptional architecture, pottery, painting, and sculpture, as well as significant advances in the fields of astronomy and mathematics. The Maya developed a system of numeration that was based on the number twenty. The basic symbols of this system are shown in the table at the right. The places in a Mayan numeral are written vertically—the bottom place represents ones, the place above represents twenties, the place above that represents 20 20, or four hundreds, and so on. For instance, this is how to write the number 997 in Mayan numerals. ••. ←. 2. 400. 800. _____ ••••. ←. 9. 20. 180. _____ •• _____ _____. ← 17. 1. . 17 997. b. If a b and c d, then ac bd.. CITY PLANNING The city of Santa Clarita requires that a parking lot can have no more than 20% of the parking spaces limited to compact cars. If a certain parking lot has 35 spaces for compact cars, how many spaces must the lot have to conform to the code? at least 175 spaces. NAME ______________________________________________ DATE. 1 m 2. a. 12 6n 12 is less than 6 times a number n.. 330. CRITICAL THINKING Give a counterexample to show that each statement is not always true.. (10) 50. 2. Use words to tell what each inequality says.. t 3. 51. ZOOS The yearly membership to the San Diego Zoo for a family with 2 adults ★ and 2 children is $144. The regular admission to the zoo is $18 for each adult and $8 for each child. How many times should such a family plan to visit the zoo in a year to make a membership less expensive than paying regular admission? at least 3 times. Chapter 6 Solving Linear Inequalities. 3 n 13 5. e. Nine is greater than or equal to one half of a quantity m. 9 . b. . 50. DRIVING Average speed is calculated by dividing distance by time. If the speed limit on the interstate is 65 miles per hour, how far can a person travel 1 1 legally in 1 hours? no more than 97 mi 2 2. 52a. Sample answer: 52. 2 3, but 4 9. 52b. Sample answer: 1 2 and 3 ★ 53. 2, but 3 4.. Reading the Lesson. a. A number n divided by 8 is greater than 5. n. 47. LONG-DISTANCE COSTS Juan’s long-distance phone company charges him 9¢ for each minute or any part of a minute. He wants to call his friend, but he does not want to spend more than $2.50 on the call. How long can he talk to his friend? no more than 27 min. 10. {kk 3}. {xx 40}. t 14 28. 48. EVENT PLANNING The Country Corner Reception Hall does not charge a rental fee as long as at least $4000 is spent on food. Shaniqua is planning a class reunion. If she has chosen a buffet that costs $28.95 per person, how many people must attend the reunion to avoid a rental fee for the hall?. Solve each inequality. Then check your solution. a 7. 14 5. ?. 45. GEOMETRY The area of a rectangle is less than 85 square feet. The length of the rectangle is 20 feet. What is the width of the rectangle? 1 less than 4 ft 4 46. FUND-RAISING The Middletown Marching Mustangs want to make at least $2000 on their annual mulch sale. The band makes $2.50 on each bag of mulch that is sold. How many bags of mulch should the band sell? at least 800 bags. 13. Half of a number is at least 14.. NAME ______________________________________________ DATE. c.. 0.40n 45; {nn 112.5}. Define a variable, write an inequality, and solve each problem. Then check your solution. 13–15. Sample answer: Let n the number.. Skills Practice, 6-2 Practice (Average). a 21 6. 40. Negative seven times a number is at least 14. 7n 14; {nn 2} 1 41. Twenty-four is at most a third of a number. 24 n; {nn 72} 3 2 42. Two thirds of a number is less than 15. n 15; {nn 22.5} 3 ★ 43. Twenty-five percent of a number is greater than or equal to 90. ★ 44. Forty percent of a number is less than or equal to 45.. Solve each inequality. Then check your solution. y 6. ?. 39. Seven times a number is greater than 28. 7n 28; {nn 4}. Exercises 1. 2. c.. Define a variable, write an inequality, and solve each problem. Then check your solution. 39–44. Sample answer: Let n the number.. Lesson 6-2. The property is also true when and are replaced with. 7, then complete each inequality.. a. a b. 4a ? 14 ? 3.5 ★ 38. If 4t 2, then complete each inequality. b. –8t ? 4 a. t ? 0.5. For all numbers a, b, and c, with c 0, Multiplication Property of Inequalities. {mm 3}; See margin for graph.. ● ●. _____ 10 _____. 1. •. • _____ 11 _____. 2. ••. •• _____ 12 _____. 3. •••. ••• _____ 13 _____. 4. ••••. •••• _____ 14 _____. 5 _____. _____ 15 _____ _____ _____ • 16 _____ _____. 0. • 6 _____ •• 7 _____ ••• 8 _____. •• _____ 17 _____ _____ _____ ••• 18 _____ _____. •••• 9 _____. •••• _____ 19 _____ _____.

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